Solution
The Lost Boarding Pass
Show the problem again
100 passengers board a plane with 100 assigned seats. The first passenger lost his ticket and sits in a uniformly random seat. Each subsequent passenger sits in their own seat if it is available, otherwise in a uniformly random empty seat. What is the probability the last passenger sits in their own assigned seat? Express as a fraction.
Worked solution
The answer is 1/2. At every point, the only seats that matter are seat #1 and seat #100. Whenever a displaced passenger chooses randomly, they are equally likely to pick seat #1 (ending the chaos in the last passenger's favor) or seat #100 (dooming them), and all other choices merely defer the same symmetric decision. By symmetry, the probability is exactly 1/2, independent of the number of passengers.
Source: Classic airline-seating probability puzzle in wide circulation; no traceable original source. Statement written for AxiomIQ.